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Last spring, my colleagues got funding for two shiny display cabinets for my
math department. Appropriate displays were just slow enough in coming that I
could (quickly) haul out several of my Lego math demonstration models and fill
an entire case with them:
Home Plate
This familiar-looking shape does not actually exist. Look at the triangle on
the bottom. (Use the dashed black line as its long side.) The side lengths of
12, 12, and 17 do not obey the Pythagorean Theorem—check 12^2+12^2 and 17^2.
However, it’s very, very nearly a right triangle, so we let it slide.
Turned Squares
Here are two squares. Look at the larger one. If you include the white dotted
lines, you can see that the larger square is made up of four right triangles
whose sides are very, very close to 12-12-17. These numbers were obtained using
a continued fraction.
The smaller square is made up of four 6-8-10 right triangles and one 2 by 2
square. The yellow and blue triangles are illustrated with dotted lines. Of
the five polygons in the display, this is the only one whose linear measurements
are all integers.
Equilateral Triangles
Here are two equilateral triangles. The triangle with sides of length 9 does
not work well in Lego. Its altitude is about 7.794—not an integer value—so its
top vertex (where yellow and red meet) is not near a grid point, so it cannot
connect to the gray baseplate.
The triangle with sides of length 15 does work well. Its altitude is about
12.99—very close to 13—so its top vertex (where blue and yellow meet) connects
solidly to the gray baseplate.
Blue Wave
This wavy object is generated by the function z=5cos(x^2+y^2)+6. In Calculus
II, we learn how to find the size of one slice; in Calculus III we learn how to
slice up the entire object and find its volume.
Attention: Dr. Masi, Dr. Masi to the fourth floor.
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